Unitary Representations and Complex Analysis

Much of what I will say depends on analogies between representation theory and linear algebra, so let me begin by recalling some ideas from linear algebra. One goal of linear algebra is to understand abstractly all possible linear transformations T of a vector space V. The simplest example of a linear transformation is multiplication by a scalar on a one-dimensional space. Spectral theory seeks to build more general transformations from this example. In the case of infinite-dimensional vector spaces, it is useful and interesting to introduce a topology on V, and to require that T be continuous. It often happens (as in the case when T is a differential operator acting on a space of functions) that there are many possible choices of V, and that choosing the right one for a particular problem can be subtle and important.